$\begingroup$Have you heard of Dedekind cuts? The idea is to identify a real number $r$ with the set $(-\infty,r)\cap\Bbb Q$. A set of that form is called a cut. Formally: A cut is a subset $A$ of $\Bbb Q$ that has no biggest element and is downward closed ($x\in A$ and $y<x$ implies $y\in A$); the reals is then defined to be the set of cuts.$\endgroup$
– Akiva WeinbergerSep 10 '15 at 21:22

$\begingroup$Also look up "[equivalence classes of] Cauchy sequences". The idea of this construction is to identify a real number with the set of all sequences of rational numbers that converge to it.$\endgroup$
– Akiva WeinbergerSep 10 '15 at 21:23

$\begingroup$Thank you, but I think Dedekind cuts are not really operational. How exactly do we 'cut' a rational number into a sum of two irrationals by using only four arithmetical operations? We need to know one of them already to find another. At least, it is useful in that way - by knowing at least one irrational number we can potentially obtain an infinite amount of them$\endgroup$
– Yuriy SSep 10 '15 at 21:27

$\begingroup$Well, the Cauchy sequence version might work better for you; it doesn't use the four basic operations, but it uses the limiting operation, which you've been implicitly using (in your infinite sums, for instance).$\endgroup$
– Akiva WeinbergerSep 10 '15 at 22:35

1 Answer
1

There are numerous constructions of the real numbers, some of which may be along the lines of what you are looking for (do note your question in not entirely clear). You may be interested in the survey article here (or its arXiv version) which surveys many constructions.

$\begingroup$Thank you for the link. Now that I think about it, it was more of a philosophical or even psychological question. I am not satisfied with the usual definitions of algebraic and transcendental numbers, so I was looking for something more concrete.$\endgroup$
– Yuriy SSep 10 '15 at 21:07

1

$\begingroup$Your paper is very nice (and you are very kind to mention my little contribution to the subject). There is a paper by F.A. Behrend "A contribution to the theory of magnitudes and the foundations of analysis" Math. Z. 63, 345-362 (1956) which I rate very highly in this connection. In particular, Behrend shows that is quite simple to define the reals using decimal expansions and simple logic to define the additive structure and prove that the additive group is complete and totally ordered. He then gets the multiplicative structure by some neat reasoning about order-preserving endomorphisms.$\endgroup$
– Rob ArthanSep 10 '15 at 21:14

$\begingroup$thanks @RobArthan I will certainly look at that article.$\endgroup$
– Ittay WeissSep 10 '15 at 21:17